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Pythagorean Comma First Attack

Pythagorean Comma First Attack
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2019-03-16 21:49
Coordinate GeometryMatrix
This example demonstrates the manifestation and impact of the Pythagorean comma in geometric figures. It involves core mathematical concepts such as matrices, coordinate geometry, trapezoids, secant lines, and general quadrilaterals. The construction highlights the use of GeoGebra commands/objects like Sequence, Circle, Rotate, Angle, Segment, Intersect, Element, CircleSector, Textfield, Point, Polygon, and Vector. It is suitable for high school and above learners to understand the practical application of complex geometric constructions and mathematical theories.

GeoGebra Commands & Parameters

  1. Sequence((3 / 2)^(k), k, 1, m) → l7
  2. Sequence((3 / 2)^(k), k, 1, n) → l5
  3. Circle((0, 0), 1) → c
  4. Sequence((cos((k * π / 6)), sin((k * π / 6))), k, 0, 11, 1) → l1
  5. Rotate(B, ((-a) * π / 180), A) → B'
  6. Angle(B', A, B) → α
  7. Segment(Element[l1, 1], Element[l1, 7]) → f
  8. Segment(Element[l1, 2], Element[l1, 8]) → g
  9. Segment(Element[l1, 3], Element[l1, 9]) → h
  10. Segment(Element[l1, 4], Element[l1, 10]) → i
  11. Segment(Element[l1, 5], Element[l1, 11]) → j
  12. Segment(Element[l1, 6], Element[l1, 12]) → k
  13. Sequence((cos(π / 2 + ((2 * π) * log(2, Element[l7, k]))), sin(π / 2 + ((2 * π) * log(2, Element[l7, k])))), k, 1, m, 1) → l6
  14. Sequence((cos(π / 2 - ((2 * π) * log(2, Element[l2, k]))), sin(π / 2 - ((2 * π) * log(2, Element[l2, k])))), k, 1, 12, 1) → l3
  15. Sequence((cos(π / 2 - ((2 * π) * log(2, Element[l5, k]))), sin(π / 2 - ((2 * π) * log(2, Element[l5, k])))), k, 1, n, 1) → l4
  16. Sequence((cos(π / 2 - ((2 * π) * log(2, Element[just5, k]))), sin(π / 2 - ((2 * π) * log(2, Element[just5, k])))), k, 1, 12, 1) → J5
  17. Intersect(f, g) → C
  18. Element(l4, prvek) → D
  19. Element(l6, prvek) → E
  20. Angle(D, C, E) → β
  21. Angle(D, C, F) → γ
  22. Element(l4, 2) → H
  23. Element(l4, 4) → J
  24. Element(l4, 6) → K
  25. Element(l4, 1) → L
  26. Element(l4, 3) → M
  27. Element(l4, 5) → I
  28. Element(l6, 5) → N
  29. Element(l6, 3) → G
  30. Element(l6, 1) → O
  31. Element(l6, 4) → P
  32. Element(l6, 2) → Q
  33. CircleSector(C, Element[l4, 1], FF) → q
  34. CircleSector(C, Element[l4, 2], Element[l4, 1]) → s
  35. CircleSector(C, Element[l4, 3], Element[l4, 2]) → t
  36. CircleSector(C, Element[l4, 4], Element[l4, 3]) → c_1
  37. CircleSector(C, Element[l4, 5], Element[l4, 4]) → d_1
  38. CircleSector(C, Element[l4, 6], Element[l4, 5]) → e_1
  39. CircleSector(C, Element[l4, 7], Element[l4, 6]) → f_1
  40. CircleSector(C, Element[l4, 8], Element[l4, 7]) → g_1
  41. CircleSector(C, Element[l4, 9], Element[l4, 8]) → h_1
  42. CircleSector(C, Element[l4, 10], Element[l4, 9]) → k_1
  43. CircleSector(C, Element[l4, 11], Element[l4, 10]) → p_1
  44. CircleSector(C, Element[l4, 12], Element[l4, 11]) → q_1
  45. CircleSector(C, FF, Element[l6, 1]) → r_1
  46. CircleSector(C, Element[l6, 1], Element[l6, 2]) → s_1
  47. CircleSector(C, Element[l6, 2], Element[l6, 3]) → t_1
  48. CircleSector(C, Element[l6, 2], Element[l6, 3]) → c_2
  49. CircleSector(C, Element[l6, 3], Element[l6, 4]) → d_2
  50. CircleSector(C, Element[l6, 4], Element[l6, 5]) → e_2
  51. CircleSector(C, Element[l6, 5], Element[l6, 6]) → f_2
  52. CircleSector(C, Element[l6, 6], Element[l6, 7]) → g_2
  53. CircleSector(C, Element[l6, 7], Element[l6, 8]) → h_2
  54. CircleSector(C, Element[l6, 8], Element[l6, 9]) → k_2
  55. CircleSector(C, Element[l6, 9], Element[l6, 10]) → p_2
  56. CircleSector(C, Element[l6, 10], Element[l6, 11]) → q_2
  57. CircleSector(C, Element[l6, 11], Element[l6, 12]) → r_2
  58. CircleSector(C, Element[l4, 12], FF) → s_2
  59. CircleSector(C, Element[l4, 7], Element[l6, 5]) → t_2
  60. CircleSector(C, Element[l4, 2], Element[l6, 10]) → c_3
  61. CircleSector(C, Element[l4, 9], Element[l6, 3]) → d_3
  62. CircleSector(C, Element[l4, 4], Element[l6, 8]) → e_3
  63. CircleSector(C, Element[l4, 11], Element[l6, 1]) → f_3
  64. CircleSector(C, Element[l4, 6], Element[l6, 6]) → g_3
  65. CircleSector(C, Element[l4, 1], Element[l6, 11]) → h_3
  66. CircleSector(C, Element[l4, 8], Element[l6, 4]) → k_3
  67. CircleSector(C, Element[l4, 3], Element[l6, 9]) → p_3
  68. CircleSector(C, Element[l4, 10], Element[l6, 2]) → q_3
  69. CircleSector(C, Element[l4, 5], Element[l6, 7]) → r_3
  70. CircleSector(C, FF, Element[l6, 12]) → s_3
  71. Circle(C, v) → t_3
  72. Intersect(t_3, i_1, 2) → T
  73. Rotate(T, (-δ), C) → T'
  74. Angle(T', C, T) → ε
  75. Rotate(T', 90°, C) → T''
  76. Textfield(δ) → pole1
  77. Textfield(m) → pole2
  78. Textfield(n) → pole3
  79. Segment(QQ, PP) → j_3
  80. Point(j_3) → U
  81. Segment(PP, OO) → i_3
  82. Polygon(HH, JJ, OO, LL, MM, II, F) → mnohoúhelník1, hh, jj, oo, ll, mm, ii, f_4
  83. Polygon(F, QQ, MM, LL, OO, GG, HH) → mnohoúhelník2, f_5, qq, j_1, l_1, n_1, gg, b_1
  84. Polygon(QQ, PP, LL, OO, GG, NN, F) → mnohoúhelník3, i_2, pp, j_2, l_2, m_2, nn, f_6
  85. Polygon(OO, LL, PP, QQ, F, HH, GG) → mnohoúhelník4, n_2, a_2, b_2, l_3, f_7, m_3, n_3
  86. Polygon(LL, MM, QQ, F, HH, JJ, OO) → mnohoúhelník5, a_3, b_3, g_4, f_8, h_4, i_4, j_4
  87. Element(l6, 6) → R
  88. Polygon(GG, NN, F, QQ, PP, R, OO) → mnohoúhelník6, k_4, l_4, f_9, m_4, n_4, r_4, p_4
  89. Polygon(HH, F, II, MM, LL, KK, JJ) → mnohoúhelník7, q_4, f_{10}, s_4, t_4, a_4, kk, b_4
  90. Textfield(δ) → pole4
  91. Segment(A, F) → c_4
  92. Segment(JJ, A) → d_4
  93. Segment(A, HH) → e_4
  94. Segment(A, OO) → g_5
  95. Segment(A, II) → h_5
  96. Segment(A, MM) → i_5
  97. Segment(A, LL) → j_5
  98. Segment(A, QQ) → k_5
  99. Segment(A, GG) → l_5
  100. Segment(A, NN) → m_5
  101. Segment(A, PP) → n_5
  102. Segment(A, KK) → p_5
  103. Segment(A, R) → q_5
  104. Segment(Q, F) → r_5
  105. Vector(S, W) → w_1
  106. Vector(Z, A_1) → u_2
  107. Segment(Z, S) → s_5
  108. Vector(B_1, C_1) → v_2
  109. Vector(D_1, E_1) → w_2

AI Drawing Prompt Reference

Draw a circle centered at the origin with radius 1. Place 12 evenly spaced points on the circle and connect them to form a regular hexagon. From one vertex of the hexagon, draw three chords connecting to other vertices on the circle. Based on matrix and coordinate geometry principles, calculate and draw the intersections of these chords with the circle. Then, create multiple sectors from these intersection points. Finally, draw smaller sectors between the larger ones to display the complex pattern formed by these geometric elements.
The prompt above was compiled from the original drawing commands and is for reference only.